In his 1918 paper on simple mirage theory,
A. WegenerAlfred Wegener introduced a useful conceptual principle that greatly simplifies the understanding of celestial mirage phenomena, such as distorted sunsets and green flashes. It's even useful in understanding astronomical refraction in general. I do not think it has been sufficiently appreciated, so I devote this page to it.
“Elementare Theorie der atmosphärischen Spiegelungen”
Annalen der Physik (4) 57, 203–230 (1918)
The principle is very simple: split the atmosphere into two parts. The upper part is essentially a normal standard atmosphere, and is responsible for the bulk of the astronomical refraction. The lower part contains the special thermal structures that give rise to mirages.
By concentrating the structures peculiar to mirages in the lower piece of the atmosphere, Wegener was able to treat them in a simplified way — for example, by largely neglecting the changes in pressure within this shallow layer, and concentrating on the thermal effects. By leaving the upper part alone, he was able to account for the great bulk of the refraction (except, of course, in ducts), without needing to do a detailed numerical integration through the whole atmosphere. Indeed, by making the lower part sufficiently thin, the refraction in the upper part could be taken to be essentially that of the whole standard atmosphere, and read from standard tables.
Wegener's own application was to distorted sunsets, and to terrestrial mirages. For the latter, the upper layer could be ignored entirely. I'd like to indicate some further uses for Wegener's principle.
De refractionibus astronomicis
Ephemerides astronomicae anni 1788: Appendix ad ephemerides Anni 1788
(Appresso Giuseppe Galeazzi, Milano, 1787) pp. 164–277
Barnaba Oriani showed that the astronomical refraction could be expanded as a series of odd powers of (tan Z), where Z is the observed zenith distance. (This series had previously been derived by Lambert in 1759). He obtained a long series expansion good out to 80 or even 85 degrees from the zenith; but out to 70° or so, just the first two terms suffice. His remarkable result was that
This expression depends on no hypothesis about either the law of heat in the atmosphere or about the density of the air at various distances from the surface of the Earth.
In other words, the effects of the atmosphere's curvature, up to the term in (tan5 Z), depend only on the temperature and pressure at the observer. In fact, the critical parameter in the coefficient of the (tan3 Z) term is just the ratio of the effective height of the atmosphere to the radius of the Earth. (This effective height is an integral known as the “height of the homogeneous atmosphere”: it is the height of a column of gas with the temperature and pressure at the observer that would weigh as much as a column of the real atmosphere.)
Oriani's theorem explains why Cassini's uniform-density model works well, except near the horizon: as the refraction from the zenith to Z.D. 70° or so does not depend on the detailed distribution of the gas, we can adopt a constant-density model as well as any other.
Of course, this convenient simplification breaks down near the horizon, where the refraction is almost entirely dependent on the structure near eye level; for observations made below the astronomical horizon, the structure below eye level is dominant. All the refraction values near the horizon depend strongly on temperature gradients in this lower region.
Oriani's result was entirely mathematical; he did not explain its physical basis. This was done, in effect, by J. B. Biot half a century later:
Sur les réfractions astronomiques
Additions a la Connaissance des Tems, 1839, pp. 3–114 (1836)
Biot showed that the angle of incidence of a ray in the upper layers is hardly influenced by the refractive structure in the lower ones, except near the horizon; instead, it depends mainly on the ratio of the height to the radius of the Earth. In particular, the ray can never be nearly horizontal in the upper layers, even if it is horizontal somewhere near the surface. Therefore, the contribution of the upper layers is almost independent of the structure in the lower ones. If we know the angle of incidence at any point in the upper region, we know the angles everywhere in it. But this angle never gets very close to 90 degrees in the upper layers.
Biot's analysis provides the physical basis for Oriani's theorem. If the lower atmosphere has some peculiar structure, its contribution to the refraction can be very different from that in the standard model. Nevertheless, the contribution of the upper layers remains almost the same. But, at small and moderate zenith distances, the lowest layers contribute little to the total, no matter what the low-level structure may be; in the region where Oriani's theorem applies, the refraction is so small that the ray is practically a straight line. Only when the lowest layers contribute greatly to the total refraction can their effect dominate; but this requires that the ray be nearly horizontal in the lowest layers.
So, near the horizon, the lowest layers can indeed influence the refraction strongly. Even so, the contribution of the upper layers hardly varies, even when the structure in the lower ones changes considerably. Here, we see the justification for Wegener's principle as well: the refraction contribution of the upper layers, even near the horizon, is practically fixed, and can be taken equal to their contribution in the standard model. Then the only variable contribution to the refraction is in the low-lying layers where the ray is nearly horizontal.
In the absence of ducting, a ray that reaches the observer can only be horizontal at or below eye level; those that are horizontal below eye level reach the observer from below the astronomical horizon. If there is a duct, a ray reaching an observer in the duct can only be horizontal within or below the duct; so the maximum altitude at which the observer can see a ray that was nearly horizontal must be near the top edge of the duct, which is never more than a few minutes of arc above the astronomical horizon. So, with or without ducts, rays that are horizontal somewhere in the atmosphere can be seen only very near the astronomical horizon or below it.
These are the only rays that can have abnormally large refraction contributions from the lower atmosphere. Consequently, the abnormal refraction must come from the air below eye level (if there is no duct), or at least from the air below the top of the duct, if there is a duct. This distinction provides a natural criterion for dividing the atmosphere into “lower” and “upper” parts, in accordance with Wegener's principle.
The magnification theorem provides another example of the principle. This can be proved by applying Wegener's principle at the height of the observer: rays at two small apparent altitudes symmetrically placed on either side of the astronomical horizon follow paths of the same shape above the observer — i.e., they suffer exactly the same refraction from the part of the atmosphere above eye level — so their difference in refraction depends only on the ray bending below eye level. The details of the argument are a bit lengthy to go into here, so they are placed on a separate page.
Incidentally, this is yet another example of how important the lowest parts of the atmosphere are at and below the horizontal: the part of the atmosphere above eye level has no influence whatever on the magnification at the astronomical horizon.
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